Physics · Atomic and quantum physics · Year 1

Many-electron atoms

How is the periodic table built, and how does an X-ray “recognise” a substance?

The situation

In a jeweller's shop the assistant holds a small “gun” against a ring — a few seconds later the proportions of gold, silver and copper appear on the screen. This device is an X-ray fluorescence analyser: it knocks out the inner electrons of atoms, and each element responds only with X-rays of its own characteristic energy — a “fingerprint”. The order of these fingerprints matches the order of the elements in Mendeleev's periodic table exactly. How are the electrons arranged in a many-electron atom, and why does each element emit its own rays?

Topic

Many-electron atoms and the periodic table. How Mendeleev's table is built: electron shells and the Pauli principle

Key ideas

Pauli exclusion principle
Ionisation energy
Moseley's law

Formulas

N_{l} = 2 (2 l + 1), N_{n} = 2 n^{2}

Pauli principle: an orbital holds at most 2 electrons, with opposite spins; a subshell $l$ holds $2 (2 l + 1)$ electrons, and a shell $n$ holds $2 n^{2}$ .

1 s 2 s 2 p 3 s 3 p 4 s 3 d 4 p 5 s 4 d \dots (n + l \to \min)

Filling order (the Madelung–Klechkovsky rule): the subshell with the smaller $n + l$ is filled first, and if these are equal, the one with the smaller $n$ .

E_{i} = \frac{13.6 eV \cdot Z_{eff}^{2}}{n^{2}} = \frac{13.6 eV}{{(n - δ)}^{2}}

Ionisation energy of a valence electron: the inner electrons screen the nucleus; $δ$ is the quantum defect.

\sqrt{f_{K α}} = \sqrt{\frac{3}{4} c R} (Z - 1), E_{K α} \approx 10.2 eV {(Z - 1)}^{2}

Moseley's law: the square root of the $K_{α}$ frequency depends linearly on the atomic number.

W_{i \to f} = \frac{2 π}{ℏ} | V_{f i} |^{2} ρ (E_{f}); Δ l = \pm 1, Δ S = 0, Δ J = 0, \pm 1

Fermi's golden rule and the selection rules for dipole transitions ( $J = 0 \to 0$ is forbidden).

Theory

Many-electron atoms and the self-consistent field

The Schrödinger equation cannot be solved exactly for an atom with two or more electrons. In the Hartree–Fock (self-consistent field) method each electron is assumed to move in the average field of the nucleus and the other electrons; the calculation is repeated until the field and the wave functions agree with each other. States are again labelled by the quantum numbers $n, l, m_{l}, m_{s}$ , but the energy now depends on $l$ as well: an $s$ electron “penetrates” close to the nucleus and is screened less, so the $2 s$ level lies below $2 p$ and $4 s$ below $3 d$ . Electrons are arranged in shells ( $n$ = 1, 2, 3… — K, L, M…) and subshells ( $s, p, d, f$ ). The atom model on the stage is not to scale: the rings only show the shells.

Pauli principle, Hund's rule and the periodic table

Pauli principle: no two electrons in an atom can have all four quantum numbers the same. Subshells are filled according to the $n + l$ rule, and the shape of the table follows from this: the s-block has 2 columns, the p-block 6, the d-block 10, the f-block 14 ( $2 (2 l + 1)$ ). Hund's rule: electrons first occupy empty orbitals one at a time with parallel spins — the total spin $S$ is a maximum (the arrows in the orbital boxes). There are exceptions too: in Cr ( $3 d^{5} 4 s^{1}$ ) and Cu ( $3 d^{10} 4 s^{1}$ ) a half-filled or completely filled $d$ subshell is more favourable. D. I. Mendeleev constructed the table in 1869 from the periodic recurrence of properties; Moseley showed in 1913 that the atomic number is the charge of the nucleus.

Ionisation energy and the alkali metals

Across a period the nuclear charge increases while the valence electrons stay in the same shell — the ionisation energy rises and is greatest for the noble gases (He 24.6 eV, Ne 21.6 eV). A new period begins with a new shell: the single $s$ electron of an alkali metal is far from the nucleus and well screened (Na 5.14 eV, Cs 3.89 eV). The “dips” within a period reveal the subshells: in B the first $2 p$ electron, and in O the first paired $2 p$ electron, is removed easily. An alkali-metal atom resembles hydrogen: the valence electron moves in the field of a “core” of inner electrons, with levels $E = - 13.6 eV / {(n - δ_{l})}^{2}$ , where $δ_{l}$ is the quantum defect (≈ 1.37 for Na $3 s$ ). The yellow line of sodium is the $3 p \to 3 s$ transition.

Helium: parahelium and orthohelium

The two electrons of helium have total spin $S = 0$ (parahelium, singlet levels ¹S, ¹P, ¹D) or $S = 1$ (orthohelium, triplet levels ³S, ³P, ³D). By the Pauli principle, in the triplet state the spatial wave function is antisymmetric — the electrons are farther apart, the repulsion is smaller, so a triplet level lies below the corresponding singlet (2³S — 19.82 eV, 2¹S — 20.62 eV). This is the exchange energy. The ground state $1 s^{2}$ can only be a singlet. The two-electron problem cannot be solved exactly: the variational method ( $Z_{eff} = 27 / 16$ ) gives −77.5 eV for the total energy, whereas experiment gives −79.0 eV. Atomic states are classified by term symbols $^{2 S + 1} L_{J}$ .

Transition probability, selection rules, LS and jj coupling

A light wave acts on an atom as a small time-dependent perturbation $V (t) = - \vec{d} \cdot \vec{E} (t)$ . First-order perturbation theory gives Fermi's golden rule: the transition rate is proportional to $| ⟨ f | \vec{d} | i ⟩ |^{2}$ . The dipole matrix element is non-zero only for $Δ l = \pm 1$ , $Δ m = 0, \pm 1$ , and the dipole does not change the spin — $Δ S = 0$ . That is why the helium spectrum consists of two independent series, and the 2³S state is metastable (lifetime about 2 hours). In light atoms LS coupling applies ( $\vec{L} = \sum {\vec{l}}_{i}$ , $\vec{S} = \sum {\vec{s}}_{i}$ , $\vec{J} = \vec{L} + \vec{S}$ ); in heavy atoms and inner shells the spin–orbit interaction is strong — jj coupling ( ${\vec{j}}_{i} = {\vec{l}}_{i} + {\vec{s}}_{i}$ ). X-ray levels are labelled in this way: L₂ = $2 p_{1 / 2}$ , L₃ = $2 p_{3 / 2}$ , which is why the $K_{α}$ line splits into the doublet $K_{α 1}$ (L₃ → K) and $K_{α 2}$ (L₂ → K); the L₁ ( $2 s$ ) → K transition ( $Δ l = 0$ ) is forbidden.

X-rays, Moseley's law and the Auger effect

In an X-ray tube the electrons striking the anode are decelerated and produce a continuous spectrum; its short-wavelength limit is $λ_{\min} = h c / e U$ . When an electron or photon knocks an electron out of the K shell, an electron falls into the vacancy from L (Kα) or from M (Kβ) — a characteristic spectrum of precisely defined energy. A second K electron screens the K electron by about one unit, so $E_{K α} \approx \frac{3}{4} \cdot 13.6 eV {(Z - 1)}^{2}$ — Moseley's law. The released energy may, instead of emerging as a photon, knock out another electron — an Auger electron. In light atoms the Auger process dominates, in heavy ones fluorescence: the fluorescence yield $ω_{K}$ is ≈ 0.04 for Al, ≈ 0.45 for Cu and ≈ 0.83 for Ag. The spectrum also contains peaks of the tube radiation scattered by the sample: the Rayleigh peak and the Compton peak with reduced energy (the Compton effect is in a separate experiment). Model: line and edge energies are values from the X-ray Data Booklet; intensities, background, detector efficiency and matrix effects are approximate.

Did you know?

Helium was found first not on Earth but in the spectrum of the Sun: during the eclipse of 1868 a yellow line (≈ 588 nm) matching no known element was observed. On Earth helium was isolated in 1895 by William Ramsay.
Henry Moseley used X-ray spectra to show that the atomic number of an element equals the charge of its nucleus, and identified the gaps in the table (43, 61, 72, 75). He was killed at the Battle of Gallipoli in 1915, aged 27.

Inside the experiment

The experiment starts from a real-life situation: you predict the result, change the controls, run the experiment and draw a conclusion. Inside are missions, theory, a quiz and a guided tour in which the camera moves and explains the key points.

More experiments in this section

The atomic nucleus and binding — Grade 11
Thermonuclear fusion — Grade 11
Detecting particles — Grade 11
Elementary particles — Grade 11
Spin, the Zeeman effect and MRI — Year 1
Bands in solids — Year 1
Schrödinger and quantum tunnelling — Year 3
Quantum dots and the potential well — Year 3
The molecule: bonding and spectra — Year 3
Superconductivity — Year 3